Learn Danish: På trods af at helten var bange, ville han udforske bjerget. - Despite the fact that the hero was afraid, he wanted to explore the mountain.

På trods af at means "despite the fact that" and is used when a full clause with a subject and a verb follows it, like helten var bange. If you only use på trods af ("despite"), you must follow it with a noun or noun phrase, such as på trods af sin frygt (despite his fear).

Can I swap på trods af at with til trods for at in this sentence?

Yes, absolutely. Both are conjunctions that mean exactly the same thing. Saying Til trods for at helten var bange... works perfectly here and doesn't change the meaning or the tone.

Why is the word order ville han instead of han ville in the second half?

This is a classic example of the Danish V2 (verb-second) rule. The entire first clause (På trods af at helten var bange) acts as the first block of information. To ensure the main verb stays in the second position of the full sentence, the verb ville has to come right after the comma, pushing the subject han after it.

Is bange an adjective, and would it change if there were multiple heroes?

Bange is an adjective meaning "afraid" or "scared", but it is an exception to the normal adjective rules because it never changes its ending. Whether it is one hero (helten var bange) or multiple heroes (heltene var bange), the word stays exactly the same.